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#### Kant’s Theory of
Geometry in Light of the Development of
Non-Euclidean Geometries

#### Martha King

With
the development of non-Euclidean geometries in the
nineteenth century, the concern arose as to whether these alternatives
constituted a refutation of Kant’s theory of geometry. Partly
the concerns were
related to Kant’s argument that geometric judgments were *a priori*
synthetic
judgments, meaning that the conclusions of geometry could not be
derived
empirically but were yet universal principles. This aspect of
universality led
some to believe that the development and subsequent proof of
non-Euclidean
geometries implied a contradiction of Kant, whose conception of
geometry was
based in Euclid.
In this article I will address whether or not Kant’s
conception of geometry can
be reconciled with the conclusions of non-Euclidean geometry, and in
what way
Euclidean and non-Euclidean geometries can be reconciled with respect
to the
sensible world.

For
Kant, geometric propositions can only be justified through the
construction of *a
priori* intuitions in the imagination, which
intuitions must of
necessity correspond with the sensible world: “[I]t follows
that the
propositions of geometry are not determinations of a mere creation of
our
poetic imagination, which could therefore not be referred with
assurance to
actual objects; but rather that they are necessarily valid of space,
and
consequently of all that may be found in space. . . .” (Prol.
287: 31).

There
are several ways in which it is thus assumed that there is no room for
non-Euclidean geometry in Kant’s theory. One entails the idea
that the
postulates of non-Euclidean geometry cannot be conceptualized *a
priori*. In addressing this concern, it is important
to note the fact
that non-Euclidean geometries have been proven to apply to space and
the
sensible world.In this
sense, if the intuitions of a non-Euclidean geometry are determined to
be *a
priori*, then there is plenty of room in Kant for
the validity of such
intuitions, provided that they in some way correspond to physical space.

There
seems to be a natural inclination to want to jump to the conclusion
that the
propositions of non-Euclidean and Euclidean geometries contradict one
another
to the extent that if they cannot be allowed to co-exist then one or
the other
must be determined to be the ‘true’ geometry of our
world. The problem is not
just that Euclidean geometry can be derived soundly from its
postulates, but
that so too can multiple non-Euclidean geometries. Since Kant was
relying on
Euclidean geometry, it is assumed that there was no room in his
epistemology
for any non-Euclidean geometry. Yet Kant’s theory on second
glance actually
seems to fit quite well with the possibility of alternate geometries.
Take, for
example, the construction of two parallel lines in my intuition.
Whether or not
my formal intuition of these two lines allows them to potentially
intersect
depends entirely on the shape which my *form* of intuition
takes—that is,
whether in this given instance I take space to be elliptical,
spherical, or
Euclidean. In each instance I will still derive what will happen with
the
parallel lines by* a
priori* intuition alone. In point of fact, Gauss,
Bolyai, and
Lobachevski “all carried out their work without recourse to
experiment, and
thus *a
priori*” (Jones 1946, p. 143). Despite
this, however, we are drawn back
to Kant’s remark in the *Prolegomena* that “the
space of the
geometer is exactly the form of sensuous intuition which we find* a
priori* in us, and contains the ground of the
possibility of all
external appearances” (288: 32). If we are to accept the
possibility that both
Euclidean and non-Euclidean geometries can be derived *a priori*, are we
‘stuck’
then when it comes to determining which one applies to external
experience?

Here we
come to several possibilities. One possibility, put forth by Paul
Henle, is to
take the division of phenomenal and physical space and argue that these
are the
“same space considered in different contexts, not of two
separate spaces”
(1962, p. 234). Another possibility, which Lucas references, is that of
Ewing
and Strawson who “have attempted to save Kant’s
account of geometry by
maintaining that it is *a
priori* true at least of phenomenal
geometry—the geometry of our visual experience—that
is, Euclidean” (Lucas 1969,
p. 6). This desire to remedy Kant’s theory of geometry with
respect to the
existence of non-Euclidean geometries by the separation of
space/geometry into
two separate realms seems entirely unnecessary. First, Kant’s
theory provides
without any difficulty the sheer*
*logical *possibility*
of alternate
geometries so long as the concepts of such geometries are not
contradictory. “A
Kantian may admit these without difficulty as being mere exercises in
deduction
having nothing to do with actual space. The physical use of
non-Euclidean
geometry is, however, another matter” (Henle 1962, p. 232).
Yet why can’t both
kinds of geometry be made manifest in the physical world? Taking light
rays to
be the physical manifestation of ‘straight lines’
can lead us, on Earth, toward
accepting Euclidean geometry under certain circumstances, but in outer
space
the investigation of black holes would lend toward the acceptance of
Riemannian
geometry. Both of these possibilities can exist in the physical
universe, and
thus it is the circumstances of the investigation at hand that call for
the
application of one or another geometry, as opposed to a strict reliance
on only
one geometry as applicable to the physical world. The obvious concern,
however,
is that the only way to determine which geometry applies in what cases
is
empirically. Nevertheless, if we are secure that the constructions of
any
geometry have been derived from *a
priori *intuitions, and that these
constructions of geometry apply or have the potential to apply to the
physical
world (including the vast reaches of outer space), then there is no
ground on
which to say that Kant’s theory excluded the possibility of
valid non-Euclidean
geometries. Nor can we say that the geometry of the universe must be
exclusively Bolyaian, Riemannian, *or *Euclidean.

Another
reason some have mistakenly assumed that non-Euclidean geometry cannot
fit
within Kant’s theory is the contention that we are incapable
of imagining (and
thus intuiting *a
priori*) any space other than Euclidean space.
This simply does not seem to be the case. I can imagine a globe in
which the
longitudinal lines intersect at the poles and yet imagine that I,
standing on
the face of the Earth looking upwards, imagine these longitudinal lines
to run
parallel over my head. In fact, Hermann von Helmholtz suggested that we
could
even “imagine ordering our perceptions in a non-Euclidean
space” by imagining
the world as reflected through a convex mirror (Grabiner 1988, p. 226).
Even
Escher’s drawings seem to suggest that it is quite possible
to imagine a world
other than the strictly Euclidean. If the figures of non-Euclidean
geometry
cannot be ‘drawn’ in the intuition, it further begs
the question as to how
Lobachevski and others were able to come up with their concepts in the
first
place. Following what Lobachevski perhaps assessed in his mind, I
can—in an
even starker example—imagine my form of intuition to be
spherical and thus
construct various geometrical possibilities within my intuition.

Thus I
contend, following Hopkins,
that we are able to both “see and picture consistently with
Euclidean and
non-Euclidean theories” (Hopkins 1973, p. 34). It is
important here to say that
while Kant, having pre-dated the development of non-Euclidean geometry,
would
have been functioning under the assumption of the possibility of
intuiting only
Euclidean concepts, I nonetheless contend that there is plenty of room
to include
the concepts of other geometries within Kant’s distinction
between formal
intuitions and the form of intuitions.

While
he did not argue specifically for the
possibility of more than one form of intuition, I would argue that
allowing for
this possibility is essential in reconciling Kant with the development
of
multiple non-Euclidean geometries and also with his own contention that
the
construction of a concept “must in its representation express
universal
validity for all possible intuitions which fall under the same
concept” (A713/B741:
577). Since both Euclidean and non-Euclidean geometries have
applicability in
both intuition and the physical world, this possibility of multiple
forms of
intuition allows Kant’s assessment of
“geometry’s unquestionable validity with
regard to all objects of the sensible world” (Prol. 292: 36)
to stretch into
the realm of non-Euclidean geometries. This indeed meets
Kant’s requirement
that the form of appearance “must allow of being considered
apart from all
sensation” (A20/B34: 66). Here I think it deserves mention
that for Kant
geometric concepts aren’t given validity superficially:
“It is, indeed, a
necessary logical condition that a concept of the possible must not
contain any
contradiction; but this is not by any means sufficient to determine the
*objective
reality *of the concept, that is, the possibility
of such an object as is
thought through the concept” (emphasis added, A220/B268:
240). Where an
impossibility crops up, however, is at the level of intuition
(A221/B269). And
thus by shifting the concept of space in the mind at the level of
intuition the
possibility of non-Euclidean geometries can be shown to be valid. Our
form of
intuition is perhaps most naturally Euclidean, as that is the geometry
we are taught and
with which we are most familiar.
However, it is clearly possible to shift our form of intuition into a
spherical
or elliptical space and work with constructions within these different
realms.
Certainly this is what Lobachevski and Bolyai must have done, for their
geometries were not entirely derived at the level of empirical
observation.
With this possibility of different forms of intuition in mind, the
axioms of
multiple geometries can thus peacefully coexist as *a priori *intuitions, but
only when the form of intuition assumes the shape in the mind which
best
corresponds to these seemingly contradictory axioms of multiple
geometries.

Jones
suggested that the reason why so many people have taken the arrival of
non-Euclidean geometries to imply a refutation of Kant is that such
people
incorrectly reason that:

[1]
Only
one geometry can correctly apply to actual space.

[2] Experience alone, therefore, can determine which geometry is true.

[3] Kant’s position that geometry is *a priori*, and independent of
experience,
is thus untenable. (Jones 1946, p. 139)

The
problem, Jones argues, is in [1]. It seems, though, that
from experience and empirical evidence we know that more than one
geometry can
apply to physical space. The question then becomes not which geometry
applies
in all cases to actual space, but rather which geometry is *necessitated*
*by*
a given instance of actual space. Thus: “Which type of
geometry proves most
suitable . . . depends on the type of lines we use, and the choice of
the type
of line for actual measurements, in turn, is affected by empirical
factors.
Regardless of the type of line, and thus of the type of geometry used,
however,
the other geometries remain sound unless it can be shown that only one
type of
line can be constructed in space” (Jones 1946, p. 143). We
see then that those
who view the development and seemingly contradictory nature of multiple
geometries as a reason to insist that only one geometry can be the
‘correct’
one “do not fully realize that the postulates of geometry are
capable of truth
or falsity only when they are interpreted in some specific way. . . .
What we
should say is that Riemannian geometry is true when, for instance, the
term
‘straight line’ is interpreted as meaning the path
of a ray of light through a
medium of uniform refractive index. . . . It is equally misleading and
false to
say simply that the postulates of Euclidean geometry are
false—for the
postulates of Euclidean geometry are true under some interpretations
and false
under others” (Barker 1964, p. 52).

For now,
we simply can’t determine *a priori* which geometry *best *describes
the sensible world. In some cases appearances may deceive us as to
which
geometry to follow, and in such cases we must rely on empirical data to
sort
out our methods. However, given that there is at least one shared
principle
among all the geometries, i.e., that they all define a
‘line’ as the “shortest
path between two points” [for example, in Euclidean geometry
this ‘line’ being
‘straight’ and in Riemannian geometry this
‘line’ being an arc] (Hopkins 1973,
p. 8), there exists the potential for a unifying theory of geometry
which would
someday incorporate all possibilities in a non-contradictory way.

Saint Louis
University

Saint Louis, Missouri

About
the Author

#### References

Barker, S.
(1964).
*Philosophy
of Mathematics*. Englewood
Cliffs, NJ: Prentice-Hall.

Grabiner,
J. V.
(1988). “The Centrality of Mathematics in the History of
Western Thought,” *Mathematics
Magazine* 61:4, 220–230.

Henle, P.
(1962).
“The *Critique
of Pure Reason* Today,” *The Journal of Philosophy* 59:9,
225–234.

Hopkins, J.
(1973). “Visual Geometry,” *The Philosophical Review* 82, 1:
3–34.

Jones, P.
C.
(1946). “Kant, Euclid,
and the Non-Euclideans,” *Philosophy of Science* 13:2,
137–143.

Kant, I. (1965). *The Critique of Pure Reason*,
trans.
Norman Kemp Smith. New York:
Bedford/
St. Martin’s.

Kant, I. (1985). *Prolegomena to Any Future Metaphysics*.
In *Philosophy
of Material Nature*, trans. James W. Ellington. Indianapolis:
Hackett.

Lucas, J.
R.
(1969). “*Euclides
ab omni naevo vindicatus*.” *British Journal for the
Philosophy of Science*, 20: 1–11.